f(x)={xsin(x1),0,x=0x=0 and g(x)=xf(x) For f(x)LHL=h→0−lim{−hsin(−h1)} =0× a finite quantity between −1 and 1 =0 RHL=h→0+limhsinh1=0 Also, f(0)=0 Thus LHL=RHL=f(0) ∴f(x) is continuous at x=0 g(x)={x2sinx1,0,x=0x=0 For g(x) LHL=h→0−lim{−h2sin(h1)} =02× a finite quantity between −1 and 1=0 RHL =h→0+limh2sin(h1)=0 Also g(0)=0 ∴g(x) is continuous at x=0
Let f, g: R → R be two functions defined by f(x)= … and g(x)=x f(x) Statement I: f is a continuous function at x=0. Statement II: g is a…
JEE Main 2014 — Mathematics Calculus
2014mcqmedium
Let f,g:R→R be two functions defined by f(x)={xsin(x1)0,,x=0,x=0 and g(x)=xf(x) Statement I: f is a continuous function at x=0. Statement II: g is a differentiable function at x=0.
Official previous-year question
Held on 12 Apr 2014 · Verified 6 Jul 2026.
Options
- A
Both statement I and II are false.
- B
Both statement I and II are true.
- C
Statement I is true, statement II is false.
- D
Statement I is false, statement II is true.
Did you get this right?
Sign in to track your attempts and accuracy.
Your note
Sign in to keep a private note on this question. Nothing you write is ever public.
JEE Main Calculus in other years
More JEE Main Calculus questions
If $\int\left(\frac{1-5 \cos ^{2} x}{\sin ^{5} x \cos ^{2} x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then $f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$ is equal to
2026hard
If the solution curve $y=f(x)$ of the differential equation $\left(x^{2}-4\right) y^{\prime}-2 x y+2 x\left(4-x^{2}\right)^{2}=0, x>2$, passes through the point $(3,15)$, then the local maximum value of $f$ is $\_\_\_\_$.
2026hard
Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_{0}^{64}\left(x^{1 / 3}-\left[x^{1 / 3}\right]\right) \mathrm{d} x$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi}\left(\frac{\sin ^{2} \theta}{\sin ^{6} \theta+\cos ^{6} \theta}\right) \mathrm{d} \theta$ is equal to $\_\_\_\_$ .
2026hard
Let $P_{1}: y=4 x^{2}$ and $P_{2}: y=x^{2}+27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y=\alpha x, \alpha>0$ and $P_{1}$, then $\alpha$ is equal to :
2026medium
If $y=y(x)$ satisfies the differential equation $16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos y \mathrm{~d} y=(1+2 \sin y) \mathrm{d} x, x>0$ and $y(256)=\frac{\pi}{2}, y(49)=\alpha$, then $2 \sin \alpha$ is equal to :
2026hard
Other Mathematics topics for JEE Main
Keep practicing Calculus
Work through every JEE Main Calculus PYQ, year by year.